We consider, for \(z\)  in the strip \(-\frac{\alpha _1+\alpha _2}{m} \le \Re (z) \le 2\) , the analytic family of distributions \(I_z\)  on
\(\mathbb {R}^{2}\)  by
\(I_z(t) = \frac{2^{z-1}}{\Gamma \left(\frac{z}{2}\right)} |t|^{z-2} \,\,\,\,\,\, \text{for} \,\, \Re (z) > 0,\) 
and by its analytic continuation for \(\Re (z) \le 0\)  (see [1]}). Let \(J_z\)  be the distribution on \(\mathbb {R}^{4}\)  defined by
\( J_z = \delta _{\mathbb {R}^{2}} \otimes I_z \,\,\,\,\,\, \text{for} \,\, -\frac{\alpha _1+\alpha _2}{m} \le \Re (z) \le 2,\) 
where \(\delta _{\mathbb {R}^{2}}\)  is the Dirac distribution on \(\mathbb {R}^{2}\)  supported at the origin. Let
\(T_z f = \widehat{(J_z \ast \mu )} \ast f\)  , for \(f \in \mathcal {S}(\mathbb {R}^{4})\) . A computation gives that
\(\Vert  J_z \ast \mu \Vert _{L^{\infty }(\mathbb {R}^{4})} \le \left| \frac{2^{z-1}}{\Gamma \left(\frac{z}{2}\right)} \right| \,\,\,\,\,\,\text{for} \,\, \Re (z) = 2,\) 
so \(\Vert  T_z f \Vert _{L^{2}(\mathbb {R}^{4})} = \left\Vert  \widehat{T_z f} \right\Vert _{L^{2}(\mathbb {R}^{4})} \le C_z\left\Vert  \widehat{f} \right\Vert _{L^{2}(\mathbb {R}^{4})} = C_z \Vert  f \Vert _{L^{2}(\mathbb {R}^{4})}\)  for \(\Re (z) = 2\) . On the other hand, one also can see that
\(\left| \widehat{J_z}(\xi ) \right| \le C \left| \xi ^{\prime \prime } \right|^{\frac{\alpha _1+\alpha _2}{m}} \,\,\,\,\,\, \text{for} \,\,\Re (z) = -\frac{\alpha _1+\alpha _2}{m}.\) 
So, Young's inequality and Theorem REF  give
\(\Vert  T_z f\Vert _{L^{\infty }(\mathbb {R}^{4})} = \Vert  (\widehat{J_z} \widehat{\mu }) \ast f \Vert _{L^{\infty }(\mathbb {R}^{4})} \le C\left| \frac{2^{1-z}}{\Gamma \left(1-\frac{z}{2}\right)} \right| \Vert  f \Vert _{L^{1}(\mathbb {R}^{4})}\) , where \(C\)  is independent of such \(z\) .
Moreover, it is easy to check, by means of Stirling formula (see e.g. [2]}), that the family \(\lbrace  T_z \rbrace \)  satisfies the hypotheses
of the Stein's complex interpolation theorem (see [3]}, pp. 205), thus
\(\Vert  T_0 \Vert _{L^{p}(\mathbb {R}^{4}) \rightarrow L^{p^{\prime }}(\mathbb {R}^{4})} < +\infty \)  for \(p= \frac{2(\alpha _1+\alpha _2+2m)}{\alpha _1+\alpha _2+4m}\) . This implies (REF ) for such \(p\)  (see [4]}, Lemma 1). Finally, it is immediate to see that
\(\mathcal {R} : L^{1}(\mathbb {R}^{4}) \rightarrow L^{2}(\Sigma , \mu )\)  is bounded, so the theorem follows from the Riesz-Thorin Convexity Theorem and
Lemma REF .
